Step by step
The question
Question: two secants P A B and P C D come from P, with PA = 6, PB = 10, PD = 12: find PC.
1 Use the rule
For two secants from one point: the outside part times the whole line is the same for both.
Why?
The two small triangles have the same angles, so they are similar.
Their sides go in the same ratio.
2 Put the numbers in
Put the numbers in.
We call PC x.
6 × 10 = x × 12
Careful!
P B is the whole secant, not only the part inside the circle.
Careful: PB is the whole line, from P to B
6 times 10 is 60.
60 = x × 12
3 Remove the 12
x is times 12.
To remove times 12, we divide both sides by 12: 5.
x = 6012 = 5
4 Check
Let's check: both sides give 60.
So it is right!
6 × 10 = 60 5 × 12 = 60
Let's look back.
First: use the rule.
Then: put the numbers in.
Then: remove the 12.
Last: check.
Answer
x = 6012 = 5